> Since at height H the ball some potential energy = mass * gravitational acceleration * Height which will be transferred into linear momentum, angular momentum, and thermal waste. Given the ball doesn't slip, it will always have the same velocity at the bottom regardless of the shape of the ramp, so the only variable is thermal waste, which you want to minimize. The way to get rid of friction is to have the ball cover the shortest distance possible from point H to point L, which is of course, a straight line. Therefore, the optimum shape is a straight ramp.

Sorry, but (1) you have the wrong answer and (2) you should check to see if the question is answered before attempting it, if it's almost 2 months old and has several replies.

Your first mistake is to assume that speed at the bottom correlates to travel time. It's the speed along the way that determines travel time, along with the arc length of the ramp. So you have to balance initial acceleration with added distance travelled.

Your second mistake is not idealizing the problem enough. This isn't one of those trick questions where minuscule air resistance or friction losses are the deciding factor.

Your third mistake is not realizing that there are no thermal losses from rolling, because the ball does not slip with respect to the ramp. This isn't a trick, I stated it in the problem. By the work-energy theorem (or just by intuition), no slipping means no displacement, which means no work, which means no energy loss.

The correct answer is a cycloid, which is the path traced out by a point on the edge of a circle as the circle rolls along a flat surface.

I think I even derived this answer once. It wasn't all that tricky, but I was majoring in physics at the time. If anyone wants to see a handwaving derivation, I can try to post one.

^v^:)^v^
FB